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Full text of "Mathematical And Physical Papers - Iii"

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ON THE MOTION OF PENDULUMS.
65
As the cylinder moves on, it carries more and more of the fluid with it, in consequence of friction. For the sake of precision, let the quantity carried by the element dl of the cylinder be defined to be that which, moving with the velocity V, would have the same momentum in the direction of the motion that is actually possessed by the elementary portion of fluid which is contained between two parallel infinite planes drawn perpendicular to the axis of the cylinder, at an interval dl, the particles composing which are moving with velocities that vary from V to zero in passing from the surface outwards. The pressure of the cylinder on the fluid continually tends to increase the quantity of fluid which it carries with it, while the friction of the fluid at a distance from the cylinder continually tends to diminish it. In the case of a sphere, these two causes eventually counteract each other, and the motion becomes uniform. But in the case of a cylinder, the increase in the quantity of fluid carried continually gains on the decrease due to the friction of the surrounding fluid, and the quantity carried increases indefinitely as the cylinder moves on. The rate at which the quantity carried is increased decreases continually, because the motion of the fluid in the neighbourhood of the cylinder becomes more and more nearly a simple motion of translation equal to that of the cylinder itself, and therefore the rate at which the quantity of fluid carried is increased would become smaller and smaller, even were no resistance offered by the surrounding fluid.
The correctness of this explanation is confirmed by the following considerations. Suppose that F(r) had been given by the equation
F(r) = Ar'1 + Br + Crl~* + Dr>
instead of (ISO), 8 being a small positive quantity. On this supposition it would have been possible to satisfy all the equations of
cylinder itself, and the tangential velocity alters very rapidly in passing from 11 ic surface outwards. At a small distance .s- from the surface of the cylinder, the rclu-tive velocity of the fluid and the cylinder, in a direction tangential to the surface, is a function of the independent variables t', s, which vanishes with  for any given value of t', however small, but which for any given value of s, however small, approaches indefinitely to the quantity determined by (131) as t vanishes. Tho communication of lateral motion is similar to the communication of temperature when the surface of a body has its temperature instantaneously raised or lowered by a finite quantity.
S. III.                                                                                5